Probing Vibrational Symmetry Effects and Nuclear Spin Economy Principles in Molecular Spin Qubits

The selection of molecular spin qubits with a long coherence time, Tm, is a central task for implementing molecule-based quantum technologies. Even if a sufficiently long Tm can be achieved through an efficient synthetic strategy and ad hoc experimental measurement procedures, many factors contributing to the loss of coherence still need to be thoroughly investigated and understood. Vibrational properties and nuclear spins of hydrogens are two of them. The former plays a paramount role, but a detailed theoretical investigation aimed at studying their effects on the spin dynamics of molecular complexes such as the benchmark phthalocyanine (Pc) is still missing, whereas the effect of the latter deserves to be examined in detail for such a class of compounds. In this work, we adopted a combined theoretical and experimental approach to investigate the relaxation properties of classical [Cu(Pc)] and a CuII complex based on the ligand tetrakis(thiadiazole)porphyrazine (H2TTDPz), characterized by a hydrogen-free molecular structure. Systematic calculations of molecular vibrations exemplify the effect of normal modes on the spin–lattice relaxation process, unveiling a different contribution to T1 depending on the symmetry of normal modes. Moreover, we observed that an appreciable Tm enhancement could be achieved by removing hydrogens from the ligand.


Synthesis of [Mg(TTDPz)(H 2 O) 2 ]·(H 2 O)·CH 3 COOH (1). [Mg(TTDPz)(H 2 O)
] was prepared according to the procedure reported in [1]. Magnesium (1.5 g, 61.6 mmol), n-propanol (60 mL) and a small chip of I 2 were introduced in a flask under a nitrogen flux. The reaction mixture was heated to 120°C and kept under stirring untill the disappearance of solid magnesium (ca 8 h). 1,2-dicyano-thiadiazole (8 g, 58.8 mmol) was then slowly added under stirring to the resulting white suspension of Mg(PrO) 2 , kept under nitrogen, and the mixture was refluxed overnight. During the reaction the colour of the mixture changes from yellow to deep blue. Unreacted 1,2-dicyano-thiadiazole and other greenish impurities were then removed by washing the mixture with CH 2 Cl 2 (200 mL); the resulting suspension was centrifugated adding i-propanol (40 mL) and acetic acid (10 mL) to separate the solid fraction, which was then washed several times with small portions of a MeOH/CH 2 Cl 2 1:2 solution. The remaining Mg(PrO) 2 and Mg(OH) 2 impurities were removed by adding 200 mL of a CH 3 COOH/H 2 O (1:1) solution to the blu-grey powder, which was then stirred until the colour changes to deep blue. The resulting deep-blue solid compound was finally filtered off and dried, affording 3.32 g 1 (5. Synthesis of H 2 TTDPz (H 2 L). Compound H 2 L was prepared according to the procedure reported in [1].
[Mg(TTDPz) (H 2 O) 2 ] · (H 2 O) · CH 3 COOH (3.0 g, 4.6 mmol) was suspended in CF 3 COOH (30 mL) and refluxed under stirring for 2 h. Water (30 mL) was then cautiously added to the stirred dispersion to quench the reaction until a blue/purple precipitate formed. The solid was filtered off, washed with water (15 mL) and acetone (2 x 5 mL) to afford compound H 2 L was obtained as a pure blue/purple powder (1.68 g, 3.0 mmol). Yield: 67%. Elemental analysis found (calcd.) for C 16  Synthesis of [Cu(TTDPz)] (1). Compound 1 was prepared according to the procedure reported in [2]. 1,2-dicyano-thiadiazole (2.1 g, 12.9 mmol) and powdered metal copper (1 g, 15.7 mmol) were added to a round bottomed Schlenk flask. The flask was sealed under a nitrogen atmosphere, and the reaction was kept at 180°C for 48 h under stirring; the colour of the mixture changed from yellow to blueThe flask was then left to cool at RT, and the resulting product was washed with CH 2 Cl 2 (3 x 100 mL) and filtered to remove unreacted 1,2-dicyano-thiadiazole and other impurities. The purple/blue filtrate was treated with an ammonia solution (30%, 3 x 100 mL) and stirred for 12 h to remove copper metal impurities. The resulting solid mixture was filtered and washed with deionized water. The obtained prussian-blue powder was dried under vacuum and purified by thermal treatment at 350°C and 10 -3 mbar, thus providing 752 mg of pure Synthesis of [Cu 0.2 Ni 0.8 (TTDPz)] (1 20% ). Compound 1 20% (75 mg, yield:75%) was obtained accordingly to the general procedure. The relative amount of the copper(II) ion was probed to be between 20% and 25% both by ICP and Magnetic measurements ( Figure S2).
Synthesis of α-[Cu 0.2 (Pc)] (2 20% ). Compound 2 20% was prepared by dissolving stoichiometric amount of Cu(Pc) and H 2 Pc (total amount 100 mg) in methanesulfonic acid (30 mL) followed by coprecipitation with a solution 3:1 H 2 O/EtOH (40 mL). The obtainment of the α-phase structure was accomplished by keeping the temperature around 0°C during all the operations [4]. After filtration, and few washing cycles with acetone and Et 2 O, the procedure yields 66 mg of 2 20% . Yield:66%. The sample crystallinity was investigated by PXRD measurements (Figure S3), and the total concentration of Cu II confirmed by dc magnetization measurements. Indeed, the χT value ( Figure S3) is the 20% of the expected one for a S = 1/2 paramagnet according to the Curie Law [5].

X-ray powder diffraction
FIGURE S10. Plot of real and imaginary components of AC susceptibility of 1 20% , χ'(cm 3 mol -1 ) and χ"(cm 3 mol -1 ) respectively, collected at different values of T and at B = 1 T(see the colormap), and reported as function of frequency (Hz). Best fits are reported in the graph as solid lines adopting the same colormap.
FIGURE S11. Plot of real and imaginary components of AC susceptibility of 1 20% , χ'(cm 3 mol -1 ) and χ"(cm 3 mol -1 ) respectively, collected at different values of T and at B = 1.6 T(see the colormap), and reported as function of frequency (Hz). Best fits are reported in the graph as solid lines adopting the same colormap.
FIGURE S12. Plot of real and imaginary components of AC susceptibility of 2 20% , χ'(cm 3 mol -1 ) and χ"(cm 3 mol -1 ) respectively, collected at different values of T and at B = 0.3 T(see the colormap), and reported as function of frequency (Hz). Best fits are reported in the graph as solid lines adopting the same colormap.
FIGURE S13. Plot of real and imaginary components of AC susceptibility of 2 20% , χ'(cm 3 mol -1 ) and χ"(cm 3 mol -1 ) respectively, collected at different values of T and at B = 1 T(see the colormap), and reported as function of frequency (Hz). Best fits are reported in the graph as solid lines adopting the same colormap.
FIGURE S14. Plot of real and imaginary components of AC susceptibility of 1 20% , χ'(cm 3 mol -1 ) and χ"(cm 3 mol -1 ) respectively, collected at different values of B and at T = 5 K(see the colormap), and reported as function of frequency (Hz). Best fits are reported in the graph as solid lines adopting the same colormap. Zero field measurement is reported in black.
FIGURE S15. Plot of real and imaginary components of AC susceptibility of 1 20% , χ'(cm 3 mol -1 ) and χ"(cm 3 mol -1 ) respectively, collected at different values of B and at T = 7.5 K(see the colormap), and reported as function of frequency (Hz). Best fits are reported in the graph as solid lines adopting the same colormap. Zero field measurement is reported in black.
FIGURE S16. Plot of real and imaginary components of AC susceptibility of 1 20% , χ'(cm 3 mol -1 ) and χ"(cm 3 mol -1 ) respectively, collected at different values of B and at T = 10 K(see the colormap), and reported as function of frequency (Hz). Best fits are reported in the graph as solid lines adopting the same colormap. Zero field measurement is reported in black.
FIGURE S17. Plot of real and imaginary components of AC susceptibility of 2 20% , χ'(cm 3 mol -1 ) and χ"(cm 3 mol -1 ) respectively, collected at different values of B and at T = 5 K(see the colormap), and reported as function of frequency (Hz). Best fits are reported in the graph as solid lines adopting the same colormap. Zero field measurement is reported in black. Pulsed EPR measurments  List of T 1 (µs) vs. T (K) extracted from fit with biexponential (T 1 slow , T 1 fast ) or stretched exponential T 1 str . A parameters are the relative amplitudes of the slowest component extracted from biexponential fit, β is the stretching exponent.   T 1 fitting details Fitting model. Expression (4) of the main text was adopted to fit the temperature dependent behaviour of T 1 . The employed expression is reported here for the sake of clarity as (S-4). The latter has been obtained by rearranging the original model reported in [9] in order to introduce our ab initio results. The main assumptions on which our model is based are: i) the trend of second derivatives magnitude involved in the model should be correlated to the first derivatives magnitude for each mode, so that the modulation on g calculated at first order should be similar to the second order modulation; ii) the modulation of A , not included in the model, is correlated to the g modulation as well; iii) the employed model is accounting for the first order g modulation contribution to the spin-phonon coupling interaction. For this reason, a scale factor must be included to encompass both the A Cu tensor modulation and higher order modulations. Additionally, the scale factor is fixed for the whole set of vibrational modes to preserve the relative weight of each mode in the relaxation process.
. Plots of experimental T 1 -1 data of 1 0.1% (top, blue circles) and 2 0.1% (bottom, red circles) reported together with the simulated curves of direct (solid lines) and Raman (dashed lines) contributions to the spin-lattice relaxation. The curves are obtained by considering theã dir andã Ram values extracted from fits and reported in Table 3 of the main text. Each Raman contribution is simulated by employing the energy of one of the most active modes which are reported in legends. Note that the normal modes with lower energy contibute to the relaxation more than high energy ones.    S4. List of calculated normal modes (cm -1 ) with their relative Spin-Phonon coupling (SPC) constants (a.u.) obtained from calculations on 2. Vibrational modes with higher SPC constants are highlighted in: blue = E g , orange = B 2g , green = B 1g , and red = A 1g , following their respective representation expressed as Mulliken notation.